Galois group: 37T4

• Label: 37T4
• Degree: $37$
• Order: $148$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: yes
• $p$-group: no
• Group: $C_{37}:C_{4}$

Group action invariants

Degree $n$:		$37$
Transitive number $t$:		$4$
Group:		$C_{37}:C_{4}$
Parity:		$-1$
Primitive:		yes
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37), (1,31,36,6)(2,25,35,12)(3,19,34,18)(4,13,33,24)(5,7,32,30)(8,26,29,11)(9,20,28,17)(10,14,27,23)(15,21,22,16)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

There are no siblings with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Label	Cycle Type	Size	Order	Representative
	$1^{37}$	$1$	$1$	$()$
	$4^{9},1$	$37$	$4$	$( 2, 7,37,32)( 3,13,36,26)( 4,19,35,20)( 5,25,34,14)( 6,31,33, 8)( 9,12,30,27) (10,18,29,21)(11,24,28,15)(16,17,23,22)$
	$4^{9},1$	$37$	$4$	$( 2,32,37, 7)( 3,26,36,13)( 4,20,35,19)( 5,14,34,25)( 6, 8,33,31)( 9,27,30,12) (10,21,29,18)(11,15,28,24)(16,22,23,17)$
	$2^{18},1$	$37$	$2$	$( 2,37)( 3,36)( 4,35)( 5,34)( 6,33)( 7,32)( 8,31)( 9,30)(10,29)(11,28)(12,27) (13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)$
	$37$	$4$	$37$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37)$
	$37$	$4$	$37$	$( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37, 2, 4, 6, 8,10,12, 14,16,18,20,22,24,26,28,30,32,34,36)$
	$37$	$4$	$37$	$( 1, 4, 7,10,13,16,19,22,25,28,31,34,37, 3, 6, 9,12,15,18,21,24,27,30,33,36, 2, 5, 8,11,14,17,20,23,26,29,32,35)$
	$37$	$4$	$37$	$( 1, 5, 9,13,17,21,25,29,33,37, 4, 8,12,16,20,24,28,32,36, 3, 7,11,15,19,23, 27,31,35, 2, 6,10,14,18,22,26,30,34)$
	$37$	$4$	$37$	$( 1, 6,11,16,21,26,31,36, 4, 9,14,19,24,29,34, 2, 7,12,17,22,27,32,37, 5,10, 15,20,25,30,35, 3, 8,13,18,23,28,33)$
	$37$	$4$	$37$	$( 1, 9,17,25,33, 4,12,20,28,36, 7,15,23,31, 2,10,18,26,34, 5,13,21,29,37, 8, 16,24,32, 3,11,19,27,35, 6,14,22,30)$
	$37$	$4$	$37$	$( 1,10,19,28,37, 9,18,27,36, 8,17,26,35, 7,16,25,34, 6,15,24,33, 5,14,23,32, 4,13,22,31, 3,12,21,30, 2,11,20,29)$
	$37$	$4$	$37$	$( 1,11,21,31, 4,14,24,34, 7,17,27,37,10,20,30, 3,13,23,33, 6,16,26,36, 9,19, 29, 2,12,22,32, 5,15,25,35, 8,18,28)$
	$37$	$4$	$37$	$( 1,16,31, 9,24, 2,17,32,10,25, 3,18,33,11,26, 4,19,34,12,27, 5,20,35,13,28, 6,21,36,14,29, 7,22,37,15,30, 8,23)$

Group invariants

Order:		$148=2^{2} \cdot 37$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		148.3
Character table:

1A

2A

4A1

4A-1

37A1

37A2

37A3

37A4

37A5

37A8

37A9

37A10

37A15

Size

1

37

37

37

4

4

4

4

4

4

4

4

4

2 P

1A

1A

2A

2A

37A10

37A9

37A5

37A8

37A2

37A3

37A15

37A4

37A1

37 P

1A

2A

4A-1

4A1

37A15

37A5

37A8

37A2

37A3

37A10

37A4

37A1

37A9

Type

148.3.1a

R

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

148.3.1b

R

$1$

$1$

$-1$

$-1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

148.3.1c1

C

$1$

$-1$

$-i$

$i$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

148.3.1c2

C

$1$

$-1$

$i$

$-i$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

$1$

148.3.4a1

R

$4$

$0$

$0$

$0$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^7$

${\zeta }_{37}^{-11}+{\zeta }_{37}^{-8}+{\zeta }_{37}^8+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{10}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6$

${\zeta }_{37}^{-17}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-4}+{\zeta }_{37}^4+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-12}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{12}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^{18}$

148.3.4a2

R

$4$

$0$

$0$

$0$

${\zeta }_{37}^{-17}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{10}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6$

${\zeta }_{37}^{-11}+{\zeta }_{37}^{-8}+{\zeta }_{37}^8+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-12}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{12}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^7$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-4}+{\zeta }_{37}^4+{\zeta }_{37}^{13}$

148.3.4a3

R

$4$

$0$

$0$

$0$

${\zeta }_{37}^{-14}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{10}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^7$

${\zeta }_{37}^{-18}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-4}+{\zeta }_{37}^4+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-11}+{\zeta }_{37}^{-8}+{\zeta }_{37}^8+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-12}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{12}$

148.3.4a4

R

$4$

$0$

$0$

$0$

${\zeta }_{37}^{-18}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6$

${\zeta }_{37}^{-17}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-12}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{12}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-4}+{\zeta }_{37}^4+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{10}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^7$

${\zeta }_{37}^{-11}+{\zeta }_{37}^{-8}+{\zeta }_{37}^8+{\zeta }_{37}^{11}$

148.3.4a5

R

$4$

$0$

$0$

$0$

${\zeta }_{37}^{-11}+{\zeta }_{37}^{-8}+{\zeta }_{37}^8+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-4}+{\zeta }_{37}^4+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^7$

${\zeta }_{37}^{-18}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{10}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-12}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{12}$

${\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6$

${\zeta }_{37}^{-17}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{17}$

148.3.4a6

R

$4$

$0$

$0$

$0$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-4}+{\zeta }_{37}^4+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-11}+{\zeta }_{37}^{-8}+{\zeta }_{37}^8+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-12}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{12}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^7$

${\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6$

${\zeta }_{37}^{-18}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{10}+{\zeta }_{37}^{14}$

148.3.4a7

R

$4$

$0$

$0$

$0$

${\zeta }_{37}^{-12}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{12}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-4}+{\zeta }_{37}^4+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6$

${\zeta }_{37}^{-11}+{\zeta }_{37}^{-8}+{\zeta }_{37}^8+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{10}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^7$

148.3.4a8

R

$4$

$0$

$0$

$0$

${\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^7$

${\zeta }_{37}^{-14}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{10}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-12}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{12}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-11}+{\zeta }_{37}^{-8}+{\zeta }_{37}^8+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-4}+{\zeta }_{37}^4+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6$

148.3.4a9

R

$4$

$0$

$0$

$0$

${\zeta }_{37}^{-6}+{\zeta }_{37}^{-1}+{\zeta }_{37}+{\zeta }_{37}^6$

${\zeta }_{37}^{-12}+{\zeta }_{37}^{-2}+{\zeta }_{37}^2+{\zeta }_{37}^{12}$

${\zeta }_{37}^{-18}+{\zeta }_{37}^{-3}+{\zeta }_{37}^3+{\zeta }_{37}^{18}$

${\zeta }_{37}^{-13}+{\zeta }_{37}^{-4}+{\zeta }_{37}^4+{\zeta }_{37}^{13}$

${\zeta }_{37}^{-7}+{\zeta }_{37}^{-5}+{\zeta }_{37}^5+{\zeta }_{37}^7$

${\zeta }_{37}^{-11}+{\zeta }_{37}^{-8}+{\zeta }_{37}^8+{\zeta }_{37}^{11}$

${\zeta }_{37}^{-17}+{\zeta }_{37}^{-9}+{\zeta }_{37}^9+{\zeta }_{37}^{17}$

${\zeta }_{37}^{-14}+{\zeta }_{37}^{-10}+{\zeta }_{37}^{10}+{\zeta }_{37}^{14}$

${\zeta }_{37}^{-16}+{\zeta }_{37}^{-15}+{\zeta }_{37}^{15}+{\zeta }_{37}^{16}$